Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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0
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3answers
26 views

evaluate the following limit else prove that limit does not exist

The function/sequence of interest is as follows: $(\frac{n!}{n!+2})^{n!}$ I have a feeling the limit does exist, as if we divide the numerator and denominator by $n!$ we get ...
3
votes
1answer
33 views

Laplace transforms of powers of cosine

During the past several hours, while studying the Laplace transform, I've started wondering what \begin{equation} \mathcal{L} \{ \cos^n(at)\}(s) \end{equation} would look like – since it won't ...
1
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0answers
15 views

Let $E$ be the set of the $n\times n$ symmetrics and $U$ the subset of those are positive definite.

Then $f(X) = X^2$ is a diffeomorphism of $U$ onto itself. Ok, first of all, I am having troubles to show that $f(U) = U.$ How can I show that the image of this function is the set $U$? For the ...
0
votes
2answers
18 views

How to show that the inverse function is $C^r, ~ r \ge 1$

Let $g : \mathbb{R}^n \to \mathbb{R}^n \in C^r(\mathbb{R}^n), ~ r\ge 1.$ Suppose that $\|Dg(x)\| < 1$ for all $x \in \mathbb{R}^n.$ Define $f(x) = x + g(x).$ I can easily show that $f(x)$ ...
0
votes
1answer
52 views

Taylor series has zero convergence radius?

Let $$f(x):=\sum_{n=0}^{\infty} \frac{f^{n}(0)x^n}{n!}$$ where the $$|f^{n}(0)| \le C\frac{\Gamma(\frac{n+1}{\alpha})}{\alpha^{\frac{n+1}{\alpha}+1}}$$ for a constant $C>0$ and $\alpha>0$. Does ...
0
votes
0answers
34 views

rational function cancellation [duplicate]

This is probably a trivial question however i cannot find the correct information online. When simplifying mappings from $\mathbb{R}$ to $\mathbb{R}$ such as: $$\frac{x(x-1)}{(x-1)}$$ Why is it ...
0
votes
1answer
48 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
2
votes
2answers
27 views

A “special” inclusion of $W^{1,p}_0(\Omega)$ in $L^{\infty}(\Omega)$

I'm in trouble with this stuff. If $\Omega$ is a bounded open set in $\mathbb{R}^n$ of class $C^1$ and $f\in W^{1,p}_0(\Omega)$, is it true or not that $f\in L^{\infty}(\Omega)$? I think so, but I ...
1
vote
1answer
54 views

Real Analysis, Folland Problem 6.2.17 Dual of $L^p$

Theorem 6.14 - Let $p$ and $q$ be conjugate exponents. Suppose that $g$ is a measurable function on $X$ such that $fg\in L^1$ for all $f$ in the space $\sum$ of simple functions that vanish outside a ...
2
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0answers
22 views

Vector Calculus: solution to Poisson equation

This is problem 8.4.17. from Marsden Vector Calculus book. Let $\rho$ be a continuous function which vanishes outside a 3D region $W$. Define ...
3
votes
1answer
43 views

Real Analysis, Folland Problem 6.1.16 $L^p$ spaces

Problem 6.1.16 - If $0 < p < 1$, the formula $\rho(f,g) = \int |f-g|^p$ defines a metric on $L^p$ that makes $L^p$ into a complete topological vector space. Attempted proof - Suppose $a,b ...
0
votes
1answer
21 views

About $L^{p}$ norms and the Hilbert transform

When we are proving $L^{p}$ estimates for the Hilbert transform, we can proceed in the following way: Step 1 Prove that $H$ maps $L^{2}$ to $L^{2}$. Step 2 Prove that $H$ maps $L^{1}$ to ...
0
votes
2answers
33 views

Proof of $ax\le bx, x>0 \Rightarrow a \le b$ using only field axioms

I'm reading ELEMENTARY CLASSICAL ANALYSIS (2nd, Marsden), 1.1.2 Proposition. How can I prove $ax\le bx \land x>0 \implies a \le b$ using only sixteen field axioms? My proof was Prove $\forall ...
2
votes
1answer
54 views

$f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$?

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$? What if $f$ is continuous on $\mathbb{R}$? I think the first question is false but ...
1
vote
1answer
20 views

For which $r$ element of $(0,\infty$) $|x|^r \to 0$ if $x \to 0$ [closed]

For which $r$ element of $(0,\infty$) $|x|^r \to 0$ if $x \to 0$? Somehow I have no idea how to solve the problem. I know that it's true for $r\geq 1$.
2
votes
2answers
40 views

Real Analysis, Folland problem 6.1.11 $L^p$ spaces

Problem 6.1.11 - If $f$ is a measurable function on $X$, define the essential range $R_f$ of $f$ to be the set of all $z\in\mathbb{C}$ such that $\{x:|f(x) - z| < \epsilon \}$ has positive ...
0
votes
1answer
53 views

How can we write (2,5) in the countable family of disjoint open intervals?

I have just read a theorem which states that "Every open subset of R is the union of countable family of disjoint open intervals". Now,I want know how can we write (2,5) in the countable family of ...
2
votes
1answer
36 views

Modulus of roots of polynomial tend to infinity

Define $f_n:\mathbb{C}\to\mathbb{C}$ and $(\alpha_n)$ such that:$$f_n(z)=\sum_{k=0}^n \frac{z^k}{k!}$$ and $f_n(\alpha_n)=0$. Prove $|\alpha_n|\to\infty$ as $n\to\infty$. I guess this makes sense ...
3
votes
1answer
29 views

When Borel functions and Baire functions are equal?

Suppose $X$ is compact metric space. Let $A$ be the smallest set of complex functions containing all continuous functions such that: If $f_n \in A$ are uniformly bounded and $f_n \to f$ pointwise ...
1
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0answers
31 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
0
votes
0answers
66 views

If $|f'(x)|\le|x|$ then $|f(x)|\le\frac{1}{2}|x|^2$ [closed]

Let $f:U\to \Bbb R^n$ ; f is $C^1$ where $U$ is an open and convex set in $R^m$ , $f$ is differentiable, $0$ belongs to $U$ , $f(0)=0$ and $\left\|f'(x)\right\| \leq \left\|x\right\| \forall x \in ...
0
votes
1answer
15 views

Laplacian on $L^2$

how does one define the laplacian on $L^2(U)$ with $U$ open set in $\mathbb{R}^n$ ? I've seen a few times that it's indeed possible, but I can figure out how. Thanks !
1
vote
1answer
73 views

Prove that $a^x$ is continuous for a>0

Here is what I need to prove: Let $a>0$ be a positive real number. Then the function $f: \Bbb R\to \Bbb R$ defined by $f(x):=a^x$ is continuous. We are not supposed to use logarithms. Some of ...
0
votes
1answer
23 views

Estimation for points in a neighbourhood of a root of a polynomial

Let $p(x)$ be a polynomial with complex coefficients and $p(\tilde x)=0$. Choose $\delta>0$ small enough, such that $\tilde x$ is the only root of $p$ in $B_\delta(\tilde x)$. I want to show that ...
2
votes
1answer
35 views

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$.

Consider the power series $\sum_{n=1}^\infty(-1)^{n-1}\frac{x^{2n+1}}{(2n+1)(2n-1)}$. Find a closed form expression for all x which converge and hence evaluate ...
0
votes
1answer
74 views

Proving $\int_{-\infty}^{+\infty} {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$

Show that $$\int_{-\infty}^\infty {{\cos(mx)}\over(x^2+a^2)(x^2+b^2)}dx={\pi(ae^{-mb}-be^{-ma})\over ab(a^2-b^2)}$$ where $a,b,m>0$ and $a$ is not equal to $b$. I already know that ...
0
votes
0answers
38 views

Are there any ways to solve equations involving logarithms, exponentials and polynomials?

Some examples: $$e^x = 2x^2 + x$$ $$\ln x = -3x -1$$ I frequently encounter with this kind of equations, in which the problems ask me to show something related to the solution. While it's unnecessary ...
0
votes
1answer
26 views

Showing that a function is not $(d,d)-$ continuous at a point.

Let $d: \mathbb R \times \mathbb R \rightarrow \mathbb R$ be a metric: $$ d(x,y) = \begin{cases} 0 & x = y \\ |x| + |y| + 3|x-y| & x \neq y \end{cases} $$ Show that the function $f: \mathbb R ...
1
vote
0answers
14 views

$\|h\|_{L^{p}} \leq C \|f\|_{L^{p}} \implies \|g\ast h \|_{L^{p}} \leq C_1 \|g\ast f\|_{L^{p}}$?

Suppose that $f, h \in L^{p}(\mathbb R) (1\leq p \leq \infty)$ so that $\|h\|_{L^{p}} \leq C \|f\|_{L^{p}}$ for some constant $C$. Take $g\in \mathcal{S}(\mathbb R^{d})$ (Schwartz Space). We note ...
-1
votes
2answers
41 views

0 as approximate eigenvalue of a matrix [closed]

i got a problem that i cant solve. And i would be grateful for some help. Given the matrix $ X = \begin{pmatrix} 0 & 1 & & \\ -1 & 0 & \ddots & \\ & \ddots & \ddots ...
0
votes
2answers
48 views

Transport equation with variable coefficients using characteristics

I want to solve the following pde: $$x\partial_xu(x,y,z)+y\partial_y(x,y,z)+\partial_zu(x,y,z)=0,(x,y,z)\in \mathbb R^3$$ $$u(x,y,0)=u_0(x,y),(x,y)\in\mathbb R^2$$ using characteristics. Until now ...
-2
votes
1answer
48 views

$f$ is a twice-differentiable function, prove there is some $x\in (-1, 1)$ such that $f '' (x) = 0$

Suppose $f: \mathbb R \to \mathbb R$ is a twice-differentiable function and that $f(-1) = -1,\; f(0) = 0$ and $f(1) = 1$. Prove that there exists some $x \in (-1, 1)$ such that $f''(x) = 0$.
1
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0answers
20 views

About the Mercer's theorem.

Doesn't the the Mercer's theorem say something stronger than just the spectral theory of compact self-adjoint operators on a Hilbert space applied to the reproducing "kernel" function? As in if I ...
2
votes
1answer
229 views

How do I prove this (limit and series)?

Let $\{a_n\}$ be a sequence of positive real numbers such that $$a_{n+1} = a_n + ca_n^2 \quad \forall n \in N_+,$$ where $c$ is a positive constant. Show that $$(1) \lim_{n \to \infty} a_n = \infty$$ ...
2
votes
2answers
74 views

Prove $\sum_{n=1}^{\infty} \frac{\sin(nx)}{1+n^3}$ is continuous and uniformly continuous on $\mathbb{R}$

Let $$f(x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{1+n^3}$$ a) Prove $f$ is continuous on $(-\infty, \infty)$ b) Is $f$ uniformly continuous? Justify your answer. I know that $f$ is the sum of ...
0
votes
0answers
25 views

Real Analysis Theorems - Help with this proof

I am having problems trying to prove the following. Given a set A = {(n,1) / n Natural and n < 6} and given F: R*R -> R, a function that belongs to C1 class and F(P) = 0 for any P inside A. I need ...
0
votes
2answers
49 views

$\int_0^\pi{d\theta\over(a-b\cos\theta)^2}={a\pi\over (a^2-b^2)^{3\over 2}}$ [closed]

Show that $$\int_0^\pi{d\theta\over(a-b\cos\theta)^2}={a\pi\over (a^2-b^2)^{3\over 2}}$$ where $a>b>0$. I'm not sure how to simplify this integral or evaluate it. Any solutions or hints are ...
1
vote
1answer
29 views

Discuss about compactness of these sets

My question is: How can I see if (in $\mathcal H=\mathcal l (\mathbb{N} )$ $B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$ ,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$ are compacts or ...
3
votes
2answers
74 views

$\int_0^\infty {dx\over (x^2+a^2)(x^2+b^2)}={\pi\over2ab(a+b)}$ [closed]

Show that $$\int_0^\infty {dx\over (x^2+a^2)(x^2+b^2)}={\pi\over2ab(a+b)}$$ where $a,b>0$. I'm not sure how to simplify this. Any solutions or hints are greatly appreciated.
0
votes
1answer
26 views

Radius of convergence of two series [duplicate]

An unproven proposition in my book states that if the series of $a_{n}z^n$ has radius of convergence $R_1$ and the series of $b_{n}z^n$ has radius $R_2$. Then the radius of convergence of ...
1
vote
1answer
24 views

Example of convergent sequence and discontinuous function

I need a counter example to show the following statement is false: A function $f$ is continuous at a point c if there exists a sequence $x_n$ such that $x_n \rightarrow c$ as $n \rightarrow \infty$ ...
-1
votes
0answers
7 views

Prove of the equivalence of semi continuity and open function

I'm totally stuck at this question. Somebody please help me~~!! Let X and y be nonempty subsets of $R^{N}$ and $R^{K}$ respectively. For a correspondence $F: X \rightarrow Y$ and $B \subset R^{K}$, ...
-1
votes
0answers
6 views

Proof of set being continuous

While I was doing my homework, I'm stuck at this question. Can somebody please explain me how this works?? Define the correspondence $B: R_{++}^{L} \times R_{++} \rightarrow R_{+}^{L}$ by $B(p,w) ...
0
votes
1answer
28 views

Proof of being a compact set [closed]

I'm trying to solve this problem but I'm really stuck and it would be nice if someone can explain me proof or any hint for this problem. Let $X \subset\mathbb R^N$ be a nonempty compact set, and $f: ...
1
vote
5answers
128 views
+50

Find real parametar $a,b,c$ such that function $f$ become convex function $f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$

Find real parametar $a,b,c$ such that function $f$ become convex function $$f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$$ My work: If $f(x)$ is convex function that means ...
1
vote
0answers
3 views

Derivate formula for Radon-transformation

For the Radon-transformation $\mathcal{R}f(r,\omega)=\int_{\{x:x\cdot\omega=r\}}f(x)\mathrm{d}\sigma(x)$ with $r\in\mathbb{R},\omega\in\mathbb{S}^{n-1}$ I want to prove the following derivative ...
0
votes
2answers
30 views

Prob. 6, Sec. 3.4 in Kreyszig's functional analysis book: The fourier coefficients minimise the distance.

Let $n \in \mathbb{N}$, let $\{ e_1, \ldots, e_n \}$ be an orthonormal set in an inner product space $X$, let $x \in X$, let $y(x) \colon= \sum_{j=1}^n \langle x, e_j \rangle e_j$, and let $z \colon= ...
7
votes
1answer
77 views

Find a Continuous Function with Cantor Set Level Sets

This was a problem from a class that I thought was really interesting. It asked to find function $f\in C[0,1]$ such that the sets $\{x:f(x)=c\}$ form a Cantor Set for all $0\leq c\leq 1$. I found a ...
0
votes
0answers
24 views

Fundamental solution for the p-harmonic and p-biharmonic equation

I am working on $p$-Laplace equation. that is $$\tag{1} -\text{div}(|\nabla u|^{p-2}\nabla u)=\delta_0 \;\; in\; \mathcal{D}'(\mathbb{R}^2), $$ and the $p$-bilaplace equation, that is $$\tag{2} ...
2
votes
1answer
60 views

Real Analysis, Folland Problem 6.1.14 $L^p$ spaces

Problem 6.1.14 - If $g\in L^{\infty}$, the operator $T$ defined by $Tf = fg$ is bounded on $L^p$ for $1\leq p \leq \infty$. Its operator norm is at most $\|g\|_{\infty}$, with equality if $\mu$ is ...